The fundamental unit of all biological life is the cell, a mass of
biomolecules in watery solution surrounded by a cell membrane. One of the characteristic
features of a living cell is that it controls the exchange of
electrically charged ions across the cell membrane and therefore the
electrical potential of its interior relative to the exterior. This
Scholarpedia page is a basic introduction into the mechanisms
underlying this process. Although most of the material applies to all biological
cells, we will focus on those mechanisms which are of particular
importance for neurons.

Cells and cell membranes

All cells are surrounded by a cell
membrane. We will neglect all the complexities of the metabolic and
structural apparatus found in the interior of the cell and simply
consider it as a little bag, formed by the cell membrane, and filled
with saline (i.e., water with ions dissolved in
it). Likewise, we will assume that the exterior of the cell is a bath
of saline. This approximation is drastic but not unreasonable, particularly since
we are here only interested in the basis of electrical information processing
within the neuron and between neurons.

The crucial element (i.e., the only one that we did not abstract
away!) is thus the cell membrane. In its simplest form, it is a
phospholipid bilayer, i.e. a layer which is only two phospholipid
molecules thick. Each of these molecules has two ends (one is a
phosphate group, the other a hydrocarbon chain, i.e. a lipid) and
these two ends have very different properties. The phosphate end is
hydrophilic (it likes to be in a watery environment and to be
surrounded by water molecules). In contrast, the lipid end is
hydrophobic (it hates to be close to water; remember that oil is
a hydrocarbon!). Love and hate for molecules means that they will
achieve a lower energy if they attain the loved state and are able
to avoid the hated states. Each molecule attempts to get into the
lowest-energy state possible.

How can a phospholipid molecule be immersed in water at one of its
ends and, at the same time, avoid to be in water at its other end? The
answer is, as so often, team work! If enough phospholipid molecules
get together, they can bundle up their oily (hydrocarbon) ends
together, forming a double-layered sheet with the hydrocarbon ends in
the center, and, at the same time, bath their phosphate ends in water
on the outside of the sheet. This does not work at the borders of
the sheet so it is best to have no ends, i.e., to close the sheet on
itself, forming a closed sphere. The result is a certain volume of
water (or saline) enclosed by a double layer of phospolipid molecules and --
Voilà! -- a cell! In fact, such artificial cells can be
made from its constituent phospolipid molecules (for references
see Scott 1975).

Conductance

From the mentioned work on artificial membranes we know that pure
phospholipid bilayers are quite good insulators (this is not
surprising: there are no free
ions in the membrane so there are no carriers to transport charges).
Their specific conductance per unit area is only about
\(g_{pure}=10^{-13}\Omega^{-1}m^{-2}\) (Goldup et al, 1970).

The conductance of biological membranes is much higher, typically
by several orders of magnitude even at rest (i.e., without
synaptic influences etc). The reason is that there are all kinds of
ion channels and other pores penetrating
the membrane and allowing additional currents to flow. It is these
currents that make cells behave in complex and interesting ways. We
will discuss some of them below.

Capacitance

According to our simplification, the inside and the outside of the
cell are both solutions of various salts in water. As opposed to the cell membrane, salt water constitutes quite a good conductor because there are free ions that can transport electrical charges.
What we have then is two conductors (the inside and the outside of the
cell), separated by an insulator (the membrane). This makes it
possible to have different amounts of electrical charges inside and
outside the cell. If we can separate a charge \(Q\) by applying an
electrical potential \(V\) across the membrane, the membrane
has by definition a
capacitance \(C=Q/V\ .\) In fact, because the membrane is so thin (only
two molecules thick, with a total thickness of about \(6\times
10^{-9}m\)), we don't need much voltage to separate the charges and
therefore the membrane capacitance is quite high; per unit area, it is
\[\tag{1}
c=\frac{C}{S} \approx 10^{-2}F m^{-2}
\]

where F is the unit of capacitance ("Farad").

The specific capacitance of biological membranes is very close to
what is obtained simply
from the dielectric constant of lipids and the thickness of the
bilayer (for a simple derivation see Hobbie, 1997)
and, unlike the conductance, the capacitance is very little
influenced by all the complexities of biology.

Electrical potentials across the membrane

Our interest is mainly on the function of neurons which is a class of
cells that uses electrical signals for information processing. How can
a cell generate such signals?
The first thing we need is some way of generating different voltages
at different parts of the system, in particular, inside and outside of
each cell. Like all cells, neurons generate this
difference by separating different ion species. More specifically,
in the cell membrane of each neuron are ion pumps, which are protein molecules that span the membrane and use metabolic energy to transport some ions inside the cell and others outside. A typical one is the \(Na^+K^+\) pump which moves
two potassium ions into the cell and, at the same time, three sodium
ions out of the cell.
After this pump has been running for some time, the
concentration of potassium inside the cell becomes larger than that
outside, and the concentration of sodium becomes larger outside than
inside. Running the pump requires energy, which is provided to the pump in the usual energy currency of the cell, the
ATP\(\rightarrow\)ADP process.

How does this generate an electric potential? Let us assume we are in
thermodynamic equilibrium which means in this context that the net
flux of ions is vanishing. (Of course, this is a dynamic
equilibrium, meaning that ions cross the membrane in both
directions but on average, the number of ions flowing in the cell is
the same as the number of ions flowing out of the cell. Therefore, the
net flux, i.e. the difference between the numbers of ions
going each direction, is zero but the numbers themselves are
not). Then, the probability \(P_{in}\) of finding a specific
ion inside the cell, as compared to the probability
\(P_{out}\) of finding it outside, depends on the energy the
ion has inside (\(E_{in}\)) vs. outside
(\(E_{out}\)). From statistical mechanics, we know that the
relation between these probabilities is given by the Boltzmann distribution:

where \(k\) is a constant called the Boltzmann factor
and \(T\) is the temperature (in Kelvin). The energy
of an ion is certainly a very complicated quantity, with all the
interactions and biochemical complexities going on in a living
system. Fortunately, the details are not important for our purposes:
We can rewrite eq. (2) in the form

and we see that only the
difference of energies counts. The biochemical milieu is not very
different inside the cell from outside the cell. Therefore, the
chemical energies of ions inside and outside the cell are about the
same, and the only real difference between the energy of the ions
inside and outside is their electrical energy. This is computed
directly as \(zeV\ ,\) where \(z\) is the valence of
the ion, \(e\) the electric unit charge (the charge of one
electron), and \(V\) the electrical potential. Therefore, all
other energy terms cancel out and we are left with

\[
\frac{P_{in}}{P_{out}}= \exp\{-\frac{ze(V_{in}-V_{out})}{kT}\}
\]

Now we can turn things around: Instead of computing the probabilities
from the voltages, we can obtain the voltages from the probabilities:
The voltage difference between the inside and the outside of the cell
is obtained as

This relation is know as the Nernst Equation. Usually, it is
expressed in terms of the ion concentrations but since the probability
of finding an ion at some location is proportional to its
concentration at this location, the formulations are equivalent. It is
also customary to set the voltage scale such that \(V_{out}=0\) (the
choice of the electric 'ground' is arbitrary; note that this
convention changed over time and that in the classical papers by
Hodgkin and Huxley which are cited below, the opposite sign is used).

We thus find that each ion species has its own voltage at which it is
in statistical equilibrium. This voltage is commonly called the
"reversal potential" of this ion because the current generated by
these ions reverses its sign when this voltage is applied to the membrane.

Membrane patch in equilibrium

In this section, we will consider a patch of cell membrane which may
contain many ion channels but which is small enough such that the
transmembrane voltage is approximately the same everywhere in the
patch. Electrically, one single ion channel is equivalent to a
resistance in series with a voltage source: The resistance
\(r_{channel}\) is simply the inverse of the conductance of
the ion channel pore (let us assume for simplicity that the channel is
permeable to a single ion species only), and the voltage
\(V_i\) is the reversal potential of the ion species
\(i\) which can pass through the channel. If we have many channels,
say \(N\ ,\) they are electrically all in parallel, so their total
resistance is \(R=r_{channel}/N\ .\) Likewise, if we have an ion channel that selectively lets different ions pass, it can be
considered as several one-ion-only channels in parallel.

Let us start by assuming that there is only one ion species, say sodium. The
reversal potential \(V_{Na}\) for sodium is computed from the
Nernst equation, eq. (4). The conductance for
sodium \(g_{Na}=R_{Na}^{-1}\) is, as just discussed, the sum of the
conductances of all channels that allow sodium ions to
pass. According to Ohm's Law,
the current across these channels is proportional to the difference in
electrical potential (voltage) across the membrane, and the
proportionality factor is just the conductance. Keeping in mind our
convention that the outside voltage is zero (ground), the current
flowing across the membrane is thus
\[\tag{5}
I_{Na}=\frac{V_{Na}}{R_{Na}}=g_{Na}V_{Na}
\]

What if there is more than one type of ion, e.g. sodium and
potassium? Currents will then flow across both types of channels
until an equilibrium is established, with the voltage inside
the cell somewhere between the reversal potentials of these two ion
species. Common names for this equilibrium potential are resting potential or leakage potential, \(V_L\ .\)

In order to compute \(V_L\ ,\) we assume that the system is
already in equilibrium, with the inside of the cell at this potential
(which is so far unknown). In order to compute the sodium current, we thus have to modify
eq. (5) as follows,

\[\tag{6}
I_{Na}=\frac{V_{Na}-V_L}{R_{Na}}=g_{Na}(V_{Na}-V_L)
\]

since the difference between the outside (at zero voltage) and the
inside (at voltage \(V_L\)) is \(V_{Na}-V_L\ .\) By
exactly the same argument, the current of \(K^+\) ions is

\[\tag{7}
I_{K}=\frac{V_{K}-V_L}{R_{K}}=g_{K}(V_{K}-V_L)
\]

where, of course, \(G_K=R_K^{-1}\) is the conductance across
the potassium channels.

We can now compute \(V_L\) from the requirement that the
system is in equilibrium. If that is the case, then conservation of
charge requires that all currents cancel each other out, or, in other
words, that the sum of all currents is zero (this is known as
Kirchhoff's Current Law),
thus \(I_{Na}+I_{K}=0\ .\) Together with
eqs. (6) and (7), we therefore
have
\[\tag{8}
g_{K} V_{K}-g_{K}V_L + g_{Na}V_{Na} -g_{Na}V_L=0
\]

The only unknown in this equation is \(V_L\) and we can solve
for it to obtain

\[\tag{9}
V_L=\frac{ g_{Na}V_{Na}+ g_{K} V_{K}} { g_{Na}+g_{K}}
\]

Of course, voltages have to be entered in this equation with their
correct signs, as they are obtained from eq. (4). In most
physiological conditions, \(V_{Na}>0\)and
\(V_K<0\ .\)

So far we have focused only on two ion species, sodium and potassium,
just as as Hodgkin and Huxley did when they developed their famous
model of the giant squid axon (Hodgkin et al, 1952; Hodgkin and
Huxley, 1952a-d). However, it can be easily seen that
equation (9) can be generalized for
more than two ion species. The resting potential is then given by the
quotient of two sums over all ion species,

\[
V_L=\frac{ \sum_i g_{i}V_{i}}{\sum_i g_{i}}
\]
where, of course, \(V_i\) is the reversal potential of ion species \(i\)
and \(g_i\) is its transmembrane conductance.
It is found that for many neurons, the resting potential is close to
\(-70mV\ .\)

We are now ready to go beyond the equilibrium state and to consider
the behavior of a cell when its transmembrane voltage changes over time.
In order to take into account the transients, we have to consider
the membrane capacitance. As mentioned, the membrane itself is a quite
good electrical isolator which means we can accumulate electrical charges on
one side that can then not cross over to the other (of course, the same
amount of charge with the opposite sign will be found on the other side).
For instance, if we maintain an inward current \( I \) for a
time \( t\ ,\) a charge of \( Q=I t \)
will accumulate on the inside. Conservation of charge requires that the rate of
charge accumulation is equal to the current,

\(
\frac{dQ}{dt}=I
\)

The voltage in a capacitor is proportional to its charge, with the constant
of proportionality being the capacity C. Since C is a constant for an ideal capacitor
(and to an excellent approximation for a cell membrane as well), we have

\(
\frac{dQ}{dt}=C \frac{dV}{dt}=I
\)

The current \(I \) in this equation is the sum of the ionic currents which we
have computed in eqs. (6) and
(7)

\(
I= g_{Na}(V_{Na}-V)+g_{K}(V_{K}-V)
\)

Combining the two previous equations, we obtain
the following Ordinary Differential
Equation for the membrane voltage:

\[\tag{10}
C\frac{dV}{dt} = g_{Na}(V_{Na}-V)+g_{K}(V_{K}-V)
\]

Note that, in equilibrium, the temporal derivative disappears and we
get equation (8) again.

For time-independent conductances (and reversal potentials), it is
customary to lump together the ionic conductances and
voltages. Indeed, we can write
eq. (10) as follows:

where \(\tau=C/g_L\) is the time constant of the cell. From this
equation it is obvious that the membrane will approach the resting
potential \(V_L\) exponentially, with a characteristic time \(\tau\ .\)

So far, we have been considering only conductances that have no voltage or time dependence.
There are many other types of conductances which play a role in neurons. One important class
of conductances results from different types of synapses which are responsible for most
of the communication between neurons. In the case of chemical synapses, the channel is closed (conductance g = 0),
until a chemical emitted from another neuron causes the channel to open (conductance > 0). In
the case of electrical synapses, there is a fixed conductance between the two coupled neurons
(conductance g > 0 always) and the current flowing between them is proportional to the difference
of the voltages in these two cells (see eq.(15) below). Another
important class of conductances are due to channels that open dependent
on the voltage of the cell itself (see below).

All of these currents have the same form as
the leakage current in eq. (13), i.e., they can all be written as

\[\tag{15}
g(V_c-V)
\]

In this equation, \(V\) is the voltage of the neuron under study and \(g\) is the conductance for the ion species (one or several) which carry the current.
\(V_c\) is a voltage which is specific for this current; it can be the reversal voltage of the
ions carried by the conductance \(g\) or, in the case of an electrical synapse (gap junction), it is
the voltage inside the partner
neuron. Note that both \(g\) and \(V_c\) can depend on time and other variables, including the
transmembrane voltage itself (see Excitability). For instance, Hodgkin-Huxley type conductances depend on the
transmembrane voltage, and they also have their own specific kinetics for opening and closing.
There can be a large number of terms of the form (15), one for each current,
and all are added to the right
hand side of eq. (13)

Interactions in space

We have so far considered the electrical activity in a patch of
membrane that is small enough (or homogeneous enough) to behave
everywhere the same. Many neurons are large (or inhomogeneous) enough
that their membranes cannot be described by a single membrane
patch. Instead, the interactions between different parts of the cell
membrane need to be taken into account. This is the topic of
Cable Theory.